Abstract

We present initial results from our ongoing program
to image the Sunyaev-Zel’dovich (SZ) effect in
galaxy clusters at 143 GHz using Bolocam;
five clusters and one blank field are described in this
manuscript.
The images have a resolution of 58 arcsec and
a radius of ≃6−7 arcmin,
which is approximately r500−2r500 for these clusters.
We effectively high-pass filter our data in order
to subtract noise sourced by atmospheric fluctuations,
but we are able to
to obtain unbiased images of the clusters
by deconvolving the effects of this filter.
The beam-smoothed RMS is ≃10μKCMB in these images;
with this sensitivity
we are able
to detect SZ signal to beyond r500
in binned radial profiles.
We have fit our images to
beta and Nagai models,
fixing spherical symmetry or allowing for
ellipticity in the plane of the sky,
and we find that the
best-fit parameter values are in general
consistent with those obtained
from other X-ray and SZ data.
Our data show no clear preference for the
Nagai model or the beta model due
to the limited spatial dynamic range of our
images.
However, our data show a definitive preference
for elliptical models over spherical models,
quantified by an F-ratio of ≃20
for the two models.
The weighted mean ellipticity of the five
clusters is ϵ=0.27±0.03,
consistent with results from X-ray data.
Additionally, we obtain model-independent
estimates of Y500, the integrated
SZ y-parameter over the cluster face to a radius of r500,
with systematics-dominated uncertainties of ≃10%.
Our Y500 values, which are free from the biases
associated with model-derived Y500 values,
scale with cluster mass in
a way that is consistent with both self-similar predictions
and expectations of a ≃10% intrinsic scatter.

Subject headings:

Galaxy clusters are the largest collapsed objects in the
universe, making them excellent tools for
studying cosmology and the astrophysics of
gravitational collapse.
They are rare excursions in the matter density
field and the formation history
of clusters is closely tied to the
composition and evolution of the universe.
As a consequence, clusters have been used
extensively to constrain cosmology.
For example, they provided the first evidence that
the matter density, Ωm,
was insufficient to close the
universe (Bahcall et al., 1997).
Additionally,
clusters provided the most
reliable information about the amplitude
of scalar perturbations, σ8,
prior to the release of WMAP results
(Viana and Liddle, 1999; Pierpaoli et al., 2001, 2003; Borgani et al., 2001; Allen et al., 2003a; Spergel et al., 2003).

Moreover, the cluster-derived measurements
of σ8
can be combined with
measurements of the normalization
of the CMB power spectrum to constrain the
total neutrino mass via the growth
of density perturbations
(Pierpaoli, 2004; Allen et al., 2003b; Vikhlinin et al., 2009).
In addition, galaxy clusters have been
forming during the same epoch that dark
energy has evolved to dominate the energy
density
the universe.
As a result, the
number of galaxy clusters as a function
of redshift is sensitive to the
dark energy density, Ωλ,
and its equation of state,
w(Haiman et al., 2001; Holder et al., 2000, 2001).
Recently, large surveys of galaxy clusters
have been completed, or have released initial
results, that constrain the
total neutrino mass and/or the properties of dark energy
(Allen et al., 2008; Mantz et al., 2010a; Vikhlinin et al., 2009; Vanderlinde et al., 2010).
Additionally, clusters are being used to test gravity
on large scales via studies of their
internal structure and distribution
in space (Schmidt et al., 2009; Diaferio and Ostorero, 2009; Martino et al., 2009; Wu et al., 2010; Moffat and Toth, 2010; Rapetti et al., 2010).

Galaxy clusters also provide
excellent laboratories for studying the
astrophysics of structure formation.
In general, galaxy clusters are
well-behaved objects, and their properties
can be predicted to fairly good precision
using simple gravitational collapse models
and self-similar scaling relations (Kaiser, 1986).
However, non-gravitational effects,
such as radiative cooling, star formation,
turbulence, magnetic field support, and
cosmic ray pressure produce deviations
from the simple gravitational models,
and the data clearly favor models that
include non-gravitational
processes (Kravtsov et al., 2006; Nagai et al., 2007).

A wide
range of observational techniques
are used to study galaxy clusters.
Optical/infrared and radio measurements
can be used to study the individual
galaxies within the cluster
(e.g., Pipino et al. (2010); Lin et al. (2009); Gralla et al. (2010));
parameters such as velocity dispersion
and richness can then be used to
understand the global properties of the
cluster(e.g., Menanteau et al. (2010a); Rines and Diaferio (2010); Serra et al. (2010); Szabo et al. (2010); Hao et al. (2010)).
Additionally, X-ray observations
are sensitive to the bremsstrahlung
emission from the hot gas in the
intra-cluster medium (ICM),
which contains ≃90% of
the baryonic mass of the cluster
(e.g., Vikhlinin et al. (2006); Gonzalez et al. (2007); Vikhlinin et al. (2009); Arnaud et al. (2009); Zhang et al. (2010)).
Galaxy clusters are also efficient gravitational
lenses, and detailed observations of the
background galaxies can be used to
determine the matter distribution within
the cluster, ≃90% of which is
in the form of dark matter (Clowe et al., 2006; Allen et al., 2008).

The ICM can also be studied using the
Sunyaev-Zel’dovich (SZ) effect (Sunyaev and Zel’dovich, 1972),
where the background CMB photons inverse
Compton scatter off of the electrons
in the ICM.
Studies of galaxy clusters using the SZ effect
are quickly maturing.
The South Pole Telescope (SPT) and
the Atacama Cosmology Telescope (ACT)
are conducting large untargeted
surveys and have already published catalogues
with dozens of clusters (Vanderlinde et al., 2010; Menanteau et al., 2010b);
they expect to detect hundreds of clusters when
the surveys are complete.
Due to the redshift independence of the SZ
surface brightness, the clusters discovered
in these surveys are, on average, at significantly
higher redshifts than those discovered with X-ray
or optical surveys.
Consequently, SZ-selected cluster catalogues
are expected to play a leading role
in further constraining cosmological
parameters such as ΩΛ and
w (e.g., Carlstrom et al. (2002)).

These large-scale SZ surveys are operating
based on the prediction that the total
integrated SZ signal over the cluster, Y,
which is proportional to the total thermal
energy of the ICM, scales in a robust way
with total cluster mass (e.g.,
Kravtsov et al. (2006)).
In practice, the integral does not generally
extend to the edge of the cluster,
which is effectively at ≳5r500,
and can be
be performed in one of
two ways:
over a spherical volume using a deprojected
SZ profile or over a cylindrical volume using
a projected SZ profile.
Most groups
have calculated the integrated SZ signal
by performing the cylindrical integration
over the cluster face within a well
defined aperture (generally r2500
or r500) (Morandi et al., 2007; Bonamente et al., 2008; Mroczkowski et al., 2009; Marrone et al., 2009; Plagge et al., 2010; Huang et al., 2010; Culverhouse et al., 2010; Andersson et al., 2010),
although a couple groups have also performed the
spherical integral (Mroczkowski et al., 2009; Andersson et al., 2010).
With the exception of Plagge et al. (2010, hereafter P10),
these groups have exclusively used parametric profiles
to describe the SZ signal in order to determine Y.
The initial scaling results from these Y
measurements are roughly consistent with the
expectation that it is a low-scatter (≃10%)
proxy for the total cluster mass.

Additionally,
several groups have made detailed SZ observations
of previously known galaxy clusters,
and these data have been used to constrain the
properties of the ICM beyond
r500(Halverson et al., 2009; Nord et al., 2009; Basu et al., 2010; Mroczkowski et al., 2009; Plagge et al., 2010).
For example, the SPT has measured radial profiles
in 15 clusters to a significant fraction of the
virial radius, and they find best-fit model
parameters that are consistent with
previous studies using X-ray data P10.
APEX-SZ has published detailed studies of
three clusters, using a joint SZ/X-ray
analysis to constrain mass-weighted temperature
profiles beyond r500 in two of these
clusters (Halverson et al., 2009; Nord et al., 2009; Basu et al., 2010).
Note that, in addition to providing complementary information
about the ICM in these previously studied
clusters, these SZ data will help constrain
and improve systematic uncertainties in
blind SZ surveys
(e.g., constraining the typical
SZ profiles and resulting detection biases
due to the cooling properties of the
central gas (Pipino and Pierpaoli, 2001)).

High resolution (≃10 arcsec) SZ measurements
are also now being made using MUSTANG (Mason et al., 2010; Korngut et al., 2010)
and SZA/CARMA (Carlstrom et al., 2002).
These observations will provide insights on the
internal structure of clusters that will be
complementary to the information obtained
via X-ray observations.
Among other applications,
these data can be used, in combination
with optical observations,
to infer the merging history and evolution
of the cluster (Croston et al., 2008).

In this work, we present Bolocam SZ observations
of five massive galaxy clusters.
We are able to measure SZ signal in our
two-dimensional images to ≃r500 and
to well beyond r500 in azimuthally averaged
radial profiles.
Using these data, we are able to constrain
the broad morphologies of these clusters
and compute observables such as
YSZ in a model-independent way.
A companion paper to this manuscript,
Ameglio et al. (2010),
presents joint X-ray/SZ deprojections
of density and temperature profiles
for these same five clusters using
Bolocam and Chandra data
(Ameglio et al., 2007).

Bolocam is a mm-wave imaging camera that operates
from the Caltech Submillimeter Observatory (CSO)
with 144 bolometric detectors covering a
circular 8 arcmin field of view (FOV) (Glenn et al., 1998; Haig et al., 2004).
For the observations described in this manuscript,
Bolocam was configured to observe at 143 GHz.
To image the clusters, we scanned the telescope
in a Lissajous pattern (Kovacs et al., 2006), where the telescope
is driven in two orthogonal directions using
sine waves with incommensurate periods.
We used an amplitude of 4 arcmin for the sinusoids,
which were oriented along the RA and dec directions
with periods of 6.28 and 8.89 sec.
These parameters were chosen to keep the FOV
on the cluster center 100% of the time while
scanning as fast as possible at the CSO (≃4 arcmin/sec)
to modulate the cluster signal above the
low-frequency atmospheric noise.
An example of the resulting
integration time per pixel is given in Figure 1.
The data were collected via 10-minute-long observations,
with the periods of the two orthogonal sinusoids
exchanged between observations.
Typically, we complete ≃100 observations
per cluster, which corresponds to approximately
50 ksec and yields a beam-smoothed RMS of
≃10μKCMB (see Table 1).

Figure 1.— Integration time per pixel, relative
to the maximum integration time,
for MS 0451.6-0305.
Our model fits include all of the
data within a circular region with
a minimum integration time of
25% of the peak integration time,
which corresponds to 6−7 arcmin
in radius.
The red box, with 10 arcmin sides,
denotes the region used for
deconvolution of the processing
transfer function.
The minimum relative integration
time within this region is also
>25%.

target

RA

dec

redshift

Bolocam time (ksec)

RMS (μKCMB)

r500 (Mpc)

Mgas,500 (M⊙)

Abell 697

08:42:58

+36:21:56

0.28

52

08.9

1.65±0.09

19.6±2.7×1013

Abell 1835

14:01:02

+02:52:42

0.25

50

08.7

1.49±0.06

14.1±1.2×1013

MS 0015.9+1609

00:18:34

+16:26:13

0.54

38

10.2

1.28±0.08

17.5±1.9×1013

MS 0451.6-0305

04:54:11

-03:00:53

0.55

53

07.7

1.45±0.12

15.6±2.2×1013

MS 1054.4-0321

10:56:59

-03:37:34

0.83

66

06.7

1.07±0.13

11.5±2.4×1013

SDS1

02:18:00

-05:00:00

-

37

09.1

-

-

Note. – A list of the clusters presented in this
manuscript.
From left to right the columns give the RA and dec of
the cluster in J2000 coordinates, the redshift of the
cluster, the amount of Bolocam integration time,
the median RMS per beam-smoothed pixel in the Bolocam map,
the radius of the cluster, and the mass of the cluster.
The values for r500 and Mgas,500 were taken
from Mantz et al. (2010b) (Abell 697 and Abell 1835) and
Ettori et al. (2009) (MS 0015.9+1609, MS 0451.6-0305,
and MS 1054.4-0321)

Table 1Cluster properties

In this paper, we present the results from five clusters
and one blank field, which are described below.
Due to the size of our resulting maps (r≃6−7 arcmin),
we have chosen to focus primarily on high-redshift
clusters, which have virial radii within the extent of our maps.
All of the clusters presented in this manuscript are
beyond z=0.25, and three of the five are
beyond z=0.50.
For reference, r500 lies within the extent of our map for a typical
1015 M⊙ cluster at z=0.25,
and at z=0.50, the virial radius lies within
our map.
Note that, throughout this work, we assume a ΛCDM
cosmology with Ωm=0.3, ΩΛ=0.7,
and H0=70 km/s/Mpc.

Abell 697

Abell 697 is a cluster undergoing a complex
merger event along the line of sight (Girardi et al., 2006).

MS 0015.9+1609
is a triaxial cluster that is elongated along the
line of sight and has an anomalously high gas
mass fraction of 27% (Piffaretti et al., 2003).

MS 0451.6-0305

MS 0451.6-0305 is a cluster that is not quite in gravitational
equilibrium, with a slightly elongated X-ray profile
(Donahue et al., 2003).

MS 1054.4-0321

MS 1054.4-0321 is
a cluster undergoing a merger, as
evidenced by the presence of two distinct sub-clumps
in the X-ray image (Jeltema et al., 2001; Jee et al., 2005).

SDS1

Our SDS1 map is centered
in the middle
of the Subaru/XMM Deep Survey (SXDS) field.
The deep XMM-Newton survey of this field reveals only
3 clusters within our map, all near the edge,
and the largest of which has a virial
mass of M200=0.8×1013 M⊙(Finoguenov et al., 2010).
Therefore, SDS1 is approximately free of signal
from the SZ effect.

In general, our data reduction followed the procedure
described in Sayers et al. (2009, hereafter S09),
with some
minor modifications.
We briefly describe the techniques below, along with
the changes relative to S09.

3.1. Calibration

Bright quasars located near the clusters were
observed for 10 minutes once every ≃90
minutes in order to determine the offset
of our focal plane relative to the telescope
pointing coordinates.
These observations were used to construct a model
of the pointing offset as a function of local
coordinates (az,el), with a single model for
each cluster.
The uncertainty in the pointing models is
≲5 arcsec.
This pointing uncertainty is quasi-negligible
for Bolocam’s 58 arcsec FWHM beams, especially
for extended objects such as clusters.
We made two 20-minute-long observations
each night of Uranus, Neptune,
or a source in Sandell (1994) for flux
calibration.
Using the quiescent detector resistance as
a proxy for detector responsivity and atmospheric
transmission,
we then fit a single
flux-calibration curve to the entire data
set.
We estimate the uncertainty in our flux
calibration to be 4.3%, with the
following breakdown:
1.7% from the
Rudy temperature model of Mars scaled to
measured WMAP values (Halverson et al., 2009; Wright, 1976; Griffin et al., 1986; Rudy et al., 1987; Muhleman and Berge, 1991; Hill et al., 2009),
1.5% in the Uranus/Neptune model referenced
to Mars (Griffin and Orton, 1993),
1.4% due to variations in atmospheric opacity (S09),
3.1% due to uncertainties in the solid angle
of our point-spread function (PSF) (S09),
and 1.5% due to measurement uncertainties (S09).

3.2. Atmospheric noise subtraction

The raw Bolocam timestreams are dominated
by noise sourced by fluctuations in the
water vapor in the atmosphere,
which have a power spectrum that rises sharply
at low frequencies.
In order to optimally subtract the atmospheric
noise, we have used a slightly modified version
of the average subtraction algorithm
described in Sayers et al. (2010, hereafter S10).
We have modified the S10 algorithm because
these cluster data contain
additional atmospheric noise caused
by the Lissajous scan pattern.
Since we are scanning the telescope parallel to
RA and dec, the airmass
we are looking
through is constantly changing.
As a result, our data contain a large amount
of atmospheric signal in narrow bands centered
on the two fundamental scan frequencies.

Following the algorithm in S10,
we first create a template of the atmosphere by
averaging the signal from all of our detectors
at each time sample (i.e., the
average signal over the FOV).
In S10, this template is subtracted from
each detector’s timestream after weighting
it by the relative gain of that detector,
which is determined from the correlation
coefficient between
the timestream and the template.
We use a single correlation coefficient
for each detector for each 10-minute-long
observation.
However, a significant fraction of the
atmospheric noise at the fundamental scan
frequencies remains in the data after application
of the S10 algorithm, indicating that
we have slightly misestimated the
correlation coefficients.
Therefore, we modified the S10 algorithm to
compute the correlation coefficients for the template
based only on the data within a narrow band
centered on the two fundamental scan frequencies.
The atmospheric noise power in these narrow
frequency bands is roughly an order of magnitude above
the broadband atmospheric noise at nearby
frequencies;
consequently, the data in these narrow bands
provide a high signal-to-noise estimate
of each detector’s response to atmospheric
signal.
The narrowband atmospheric noise features
are completely removed using this modified
S10 algorithm, and the amount of residual
broadband atmospheric noise is slightly
reduced compared to the results
from the original S10 algorithm.

Figure 2.— Timestream noise PSD for a typical
Bolocam detector. The black curve shows the
raw PSD recorded by the detector;
spectral lines at the fundamental scan frequencies
are clearly seen above the broadband
atmospheric noise.
The red curve shows the noise PSD after
subtracting the atmospheric noise
using the average signal over the FOV.
This timestream is then high-pass filtered
at 250 mHz to produce the green PSD.
Note that there is very little cluster
signal above ≃2 Hz, where there
are some spectral lines due to
the readout
electronics.
The dashed horizontal line provides an
estimate of the photon, or BLIP,
noise.

After applying this average subtraction
algorithm to the timestream data, we then
high-pass filter the data according to

F=1−11+(10f/f0−1)κ

with f0=250 mHz and κ=8.
The value of f0 was chosen to maximize
the spatially-extended S/N for the typical cluster in our
sample based on tests with
f0 varying from
0 to 400 mHz,
and the value of κ was chosen to
produce a sharp cutoff with minimal ringing.
Figure 2 shows a typical pre and
post-subtraction timestream noise PSD.

3.3. Transfer function of the atmospheric noise filtering

In addition to subtracting atmospheric noise,
the FOV-average subtraction and timestream
high-pass filter also remove some cluster signal.
Since we use the data timestreams to both determine
the atmospheric fluctuation template and the
correlation coefficient of each detector’s data
timestream with the template, the FOV-average
subtraction acts on the data in a non-linear way.
Consequently,
its impact on the data depends on
the cluster signal.
As described below, we quantify the effects
of the FOV-average subtraction and the
timestream high-pass filter via simulation by
processing a known cluster image through
our data-reduction pipeline.
These simulations are computation-time
intensive, and
we find, in practice (see Section 6.1),
that the filtering is only mildly dependent
on the cluster signal.
Thus, in the end, we determine the
effects of the filtering for a particular
cluster’s data set using the cluster
model that best fits those data.
We use the term transfer function to describe
the effect of the filtering,
although this terminology is not rigorously
correct because the filtering depends
in the cluster signal.

To compute the transfer function, we first insert
a simulated, beam-smoothed cluster profile into our
data timestreams by reverse mapping it
using our pointing information.
These data are then processed in an identical
way to the original data, and an output image,
or map, is produced.
When processing the data-plus-simulated-cluster
timestreams we use the FOV-average subtraction
correlation coefficients that were determined
for the original data.
This ensures that the simulated cluster is
processed in an identical way to the real
cluster in our data.
In the limit that the best-fit cluster model
is an accurate description of the data,
this process is rigorously correct.
The original data map is then subtracted from
this data-plus-simulated-cluster map to
produce a noise-free image of the processed
cluster.
In Figure 3, we show an example
cluster image, along with the
noise-free processed image of the same cluster.
The Fourier transform of this
processed cluster image is divided by the Fourier transform
of the input cluster to determine how
the cluster is filtered as a function
of 2-dimensional Fourier mode (i.e.,
what we term the
transfer function, see Figure 4).

Figure 3.— Images of the best-fit spherical Nagai model
for MS 0451.6-0305.
The left image is the model and the right image
is the model after being processed through our
data reduction pipeline,
which high-pass filters the image in a
complex way.
This filtering significantly reduces
the peak decrement of the cluster and
creates a ring of positive flux
at r≳2 arcmin.
Note that the processed image is not
quite azimuthally symmetric.Figure 4.— The magnitude of the transfer
function for MS 0451.6-0305 as a
function of Fourier wavenumber u=1/λ.
At large scales, or small u,
the measurement error is negligible and
the error bars provide an indication
of the azimuthal variation.
At u>0.75 arcmin−1, the measurement
error becomes non-negligible and
we set the transfer function equal
to 1.
Note that this azimuthally averaged
transfer function is for display purposes only;
we have used the full two-dimensional transfer
function throughout our analysis.

At small angular scales, there is very little
signal in the beam-smoothed input cluster,
and numerical noise prevents us from
accurately characterizing the transfer
function at these scales.
The transfer function is expected to be
unity at small angular scales, and is asymptoting
to this value at the larger scales where
we can accurately characterize it.
Therefore, we set the transfer function to
a value of 1 for u>0.75 arcmin−1.
Deconvolving the processed cluster image
using the transfer function
(see Section 5), which
has been approximated as 1 at small angular
scales, produces an image that is slightly
biased compared to the input cluster.
The residuals between these two images are
approximately white, with an RMS of ≲0.1μKCMB.
This transfer-function-induced bias
is negligible compared to our
noise, which has an RMS of ≃10μKCMB.

As noted above, the transfer function (weakly) depends
on the
profile of the cluster;
larger clusters are more heavily filtered than
smaller clusters.
Therefore, we determine a unique transfer
function for each cluster using the
best-fit elliptical Nagai model
for that cluster (Nagai et al., 2007, hereafter N07).
The details of this fit are given in
Section 4.
Since the transfer function depends
on the best-fit model, and vice versa,
we determine the best-fit model
and transfer function in an iterative way.
Starting with a generic cluster profile, we
first determine a transfer function,
and then fit an
elliptical Nagai model using this transfer function
(i.e., the Nagai model parameters
are varied while the transfer function is
held fixed).
This process is repeated, using the
best-fit model from the previous iteration
to calculate the transfer function,
until the best-fit model parameters
stabilize.
This process converges fairly quickly,
usually after a single iteration
for the clusters in our sample.
The model dependence of the resulting
transfer function is quantified in
Section 6.1.

3.4. Noise estimation

In order to accurately characterize the
sensitivity of our images,
we compute our map-space noise
directly from the data via 1000
jackknife realizations of our cluster images.
In each realization, random subsets of half
of the ≃100 observations
are multiplied by −1 prior
to adding them into the map.
Each jackknife preserves the noise properties
of the map while removing all of the astronomical
signal, along with any possible fixed-pattern
or scan-synchronous noise due to the
telescope scanning motion6.
Since these jackknife realizations remove
all astronomical signal, we estimate
the amount of astronomical noise in our images
separately, as described below.
After normalizing the noise estimate of each
map pixel in each jackknife by the square root
of the integration time in that pixel,
we construct a sensitivity histogram, in μKCMB-s1/2,
from the ensemble of map pixels in
all 1000 jackknifes.
The width of this histogram provides an
accurate estimate of our map-space sensitivity
(see Figure 5).
We then assume that the noise covariance matrix
is diagonal7
and divide by the square root of the integration time in each
map pixel to determine the noise RMS in
that pixel.
This method is analogous to the one used in
S09, where it is
described in more detail.

There is a non-negligible amount of noise
in our maps from two main types of astronomical
sources:
anisotropies in the CMB and unresolved point
sources.
The South Pole Telescope (SPT) has recently published
power spectra for both of these sources
at 150 GHz
over the range of angular scales probed
by our maps, so we use their measurements
to estimate the astronomical noise in our
maps (Lueker et al., 2010; Hall et al., 2010).
Note that these measurements inherently include all
astrophysical effects, such as lensing and
clustering of point sources.
Using the SPT power spectra,
we generate simulated maps of our cluster
fields assuming Gaussian fluctuations.
Note that this is a poor assumption for both the SZ-sourced
CMB fluctuations and the signal sourced
by background galaxies that are lensed by the cluster.
However, since the noise power from both of these
sources is quasi-negligible compared
to the total noise in our images,
any failure of our Gaussian assumption
will have a minimal impact on our results.
The good match of the
SDS1 data to our noise model validates
our Gaussian assumption (see Figure 5).
These simulated images of the CMB plus point sources
are then reverse mapped
into our timestream data and processed to
estimate how they will appear in our
cluster maps.
We then add these processed astronomical
realizations to our jackknife realizations
to provide a complete estimate of the noise
in our images.
Sensitivity estimates from our signal-free
map of SDS1 agree well with the sensitivity
estimates from our noise maps,
indicating that there are no additional
noise sources that have not been included
in our estimate
(see Figure 5).
The properties of this noise are described
in more detail in Section 6.2.
In general, the noise from astronomical
sources is quasi-negligible compared
to the other noise in our images.
However, the large-scale correlations
from CMB fluctuations are non-negligible
in our deconvolved images (see Table 5).

Figure 5.— Histogram of the per-pixel sensitivity
for our maps of the blank-field SDS1.
The black line shows the average histogram
for each of the 1000 jackknife realizations
of the data, with noise from the CMB and
unresolved point
sources added as described in Section 3.4.
The sensitivities derived from the
noise realizations are well described by a
Gaussian fit, with the fit quality
quantified by a PTE of 0.44.
Overlaid in red is the histogram for our map
of SDS1, which is also well described
by the Gaussian fit to the noise
realizations (PTE = 0.63),
indicating that our noise model
adequately describes the data.
The dashed blue line shows the best-fit
Gaussian to the noise realizations and
has an RMS of 4.02 μKCMB-s1/2.

Note that we have chosen to
model the signal
from point sources as an additional noise
term in our maps
rather than attempting to subtract
any individual point sources.
Part of the motivation for this approach
is the fact that we do not detect any point
sources in our cluster or blank-field images.
For reference, P10 mapped 15 clusters
to a similar depth using the SPT and only
detected two point sources;
the combined area of the SPT images is approximately
20 times the combined area of our 6 images.
Although there are several known SMGs and AGNs
in our images (Zemcov et al., 2007; Ivison et al., 2000; Cooray et al., 1998),
even the brightest sources
will have a flux of ≃10μKCMB in
our observing band,
rendering them undetectable
given our noise RMS of ≃10μKCMB.
Additionally, it is not possible to
reliably estimate
the flux of these sources in our
observing band since most have only been
detected at one or two wavelengths,
generally separated from our
observing band by more than a factor of 2 in
wavelength.

4.1. The SZ effect

As mentioned above, the SZ effect involves
CMB photons inverse Compton scattering
off of hot electrons in the ICM.
Since the electrons are many orders of
magnitude hotter than the CMB photons,
there is, on average, a net increase in the energy
of the photons.
The classical distortion relative to the blackbody
spectrum of the CMB is given by

f(x)=xex+1ex−1−4,

where x=hν/kBTCMB.
The magnitude of the distortion is
proportional to the product of the
density and temperature of the electrons in the ICM
projected along the line of sight.
The frequency-dependent temperature change
of
the CMB is given by

TSZ=f(x)yTCMB,

where

y=∫neσTkBTemec2,

and ne and Te are the density
and temperature of the ICM electrons.
Relativistic corrections can be included
by multiplying f(x) by
(1+δ(x,Te))(Itoh et al., 1998).
Note that the X-ray brightness of the ICM
is proportional to n2eT1/2e;
the differing density and temperature
dependence of the SZ and X-ray signals
makes them
highly complementary probes of the
ICM (e.g., Bonamente et al. (2006), hereafter B06).

4.2. Models

We have fit our data to
two types of models: an isothermal beta
model (Cavaliere and Fusco-Femiano, 1976, 1978)
and the pressure profile proposed by N07,
hereafter the Nagai profile.
The isothermal beta model has been used extensively
to describe X-ray and SZ measurements of the
ICM, and our beta model fit parameters
can be directly compared to these
previous results.
However, the beta model provides a poor
description of deep X-ray data,
which generally have a cuspier core and
steeper outer profile
(e.g., Vikhlinin et al. (2006), N07, and Arnaud et al. (2009)).
In contrast, the Nagai pressure profile,
which is a generalization
of the NFW dark matter profile, is able to
describe deep X-ray observations over a wide
range of scales (N07, Arnaud et al. (2009)).
Therefore, we have also fit our data
to the Nagai model.
In practice, the beta and Nagai
models are highly degenerate over the
range of angular scales to which our
data are sensitive (1−12 arcmin).

Specifically, the isothermal beta model is described by

p=p0(1+r2/r2c)3β/2,

where p is the pressure profile, p0 is the
pressure normalization, rc is the core
radius, and β is the power law slope.
The beta model can be analytically integrated along
the line of sight to give the observed SZ signal,
with

TSZ=f(x)y0TCMB(1+r2/r2c)(3β−1)/2+δT,

where TSZ is the SZ signal in our map (in μKCMB)
and y0 is the central Comptonization.
We fix the value of β at 0.86 for all of our
fits;
this is the best-fit value found in P10
for SZ data8.
Since our data timestreams are high-pass filtered, we
are not sensitive to the DC signal level
in our images.
Note that because the timestream data, rather than the map,
are high-pass filtered,
the DC signal level of the map is in general
not equal to 0.
However, the DC signal of the map
is not physically
meaningful due to the filtering and must therefore
be included as a free parameter, δT,
in the model fits.
We have also generalized the beta model to
be elliptical in the plane of the sky with

TSZ=f(x)y0TCMB(1+r21r2c+r22(1−ϵ)2r2c)(3β−1)/2+δT,

where r1 is oriented along the major axis,
described by a position angle of θ
(in degrees east of north),
r2 is orthogonal to the major axis,
and ϵ is the ellipticity.

The Nagai model is described by

p=p0(r/rs)C[1+(r/rs)C](B−C)/A,

(1)

where p is the pressure,
p0 is the pressure normalization, rs is the
scale radius (rs=r500/c, with
c≃1.2(Arnaud et al., 2009)),
and A, B, and C
are the power law slopes at intermediate, large,
and small radii compared to rs.
We have fixed the values of A, B,
and C to the best-fit values found
in Arnaud et al. (2009) (1.05,5.49,0.31).
The Nagai model is not analytically integrable,
so we numerically integrate p along the line of
sight to determine TSZ.
We have also generalized the Nagai model to be
elliptical in the plane of the sky,
using the same notation as our elliptical
generalization of the beta model.
Note that, in all of our fits, we have corrected
for relativistic effects via the approximations
given in Itoh et al. (1998) using gas temperature
estimates from X-ray observations of these
clusters (Cavagnolo et al., 2008; Jeltema et al., 2001).
The relativistic
corrections are ≃5% for the typical
temperatures in these massive clusters (≃10 keV).

We find that both the beta model and the Nagai
model adequately describe our data,
with the exception of Abell 697,
and neither model is preferred with
any significance.
However, the elliptical models (χ2/DOF =5605/5413)
provide a much better fit to the full ensemble
of our cluster data compared to the
spherical models (χ2/DOF =5813/5423,
see Tables 2 and 3).
The F-ratio for these two fits is 20.1 for
ν1=10 and ν2=5413 degrees of freedom
(Bevington and Robinson, 1992),
corresponding to a probability of <10−36
that we would obtain data with equal or greater preference
for elliptical models if the clusters were spherical.
If we neglect Abell 697, which is not well described by
either model, then the F-ratio is 12.2 with
a corresponding probability of <10−16.
The weighted mean ellipticity of the five
clusters is ϵ=0.27±0.03,
consistent with results from X-ray data
(e.g., Maughan et al. (2008); De Filippis et al. (2005)).
Our inability to distinguish between the Nagai
and beta models is likely due to the limited
spatial dynamic range of our images
(≃1−12 arcmin).
For the clusters in our sample, we are insensitive
to the core, r≲0.15r500≃0.5 arcmin, and the
outskirts of the cluster r≳2r500≃10 arcmin,
which means we are only sensitive to a single power law
exponent in the Nagai model, A.
For this reason, the Nagai model is highly degenerate
with the beta model for the angular scales probed
by our data.

cluster

p0 (10−11ergcm3)

rs (arcmin)

c500

ϵ

θ (deg)

χ2/DOF

PTEχ2

PTEsim

elliptical Nagai model

Abell 697

09.3±1.3

6.9±1.0

0.93±0.14

0.37±0.05

−24±4

1289/1117

0.00

0.00

Abell 1835

08.1±1.7

6.7±1.5

0.94±0.21

0.27±0.07

0−16±10

966/945

0.31

0.22

MS 0015.9+1609

06.7±1.4

5.5±1.1

0.62±0.13

0.24±0.08

+068±12

1079/1117

0.79

0.73

MS 0451.6-0305

08.7±1.6

4.7±0.7

0.80±0.14

0.26±0.06

+85±7

1188/1117

0.07

0.08

MS 1054.4-0321

06.5±1.9

3.6±1.0

0.64±0.19

0.09±0.07

00−1±32

1084/1117

0.75

0.72

spherical Nagai model

Abell 697

11.0±1.8

4.6±0.8

1.40±0.22

-

-

1399/1119

0.00

0.00

Abell 1835

08.9±2.1

5.1±1.2

1.24±0.27

-

-

1007/947

0.08

0.07

MS 0015.9+1609

06.0±1.1

5.4±1.1

0.63±0.12

-

-

1100/1119

0.66

0.61

MS 0451.6-0305

10.6±2.1

3.5±0.5

1.08±0.15

-

-

1220/1119

0.02

0.02

MS 1054.4-0321

06.0±1.8

3.6±1.0

0.64±0.16

-

-

1087/1119

0.75

0.75

Note. – Table of the best-fit parameters and 1σ
uncertainties for our Nagai model fits
with the power-law exponents fixed to the best-fit values found
in Arnaud et al. (2009) (1.05,5.49,0.31).
From left to right the columns give the normalization
of the Nagai profile, p0, the scale radius of the major axis, rs,
the concentration parameter, c500,
the ellipticity, ϵ, the position angle of the
major axis in degrees east of north, θ,
the χ2 and DOF for the fit, and the
goodness of fit quantified by the probability to
exceed the given χ2/DOF based on the
standard χ2 probability distribution function
and also
empirically by the fraction of our 1000
noise realizations that produce a larger
χ2 value when a model cluster is
added to them
(see Section 6.2).
The values of c500 for the spherical
fits can be compared to
the nominal value of 1.17
that is given in Arnaud et al. (2009)
based on spherical fits of X-ray data.
See Section 6.2 for a description
of how the parameter uncertainties are calculated.
Note that we have included the 4.3% uncertainty
in our flux calibration in the error estimates for
p0.

Table 2Nagai model fit parameters

cluster

y0 (10−4)

rc (arcsec)

rc/r500

ϵ

θ (deg)

χ2/DOF

PTEχ2

PTEsim

elliptical beta model

Abell 697

2.93±0.26

098±10

0.26±0.03

0.38±0.04

−10±2

1288/1117

0.00

0.00

Abell 1835

2.49±0.31

079±11

0.21±0.03

0.28±0.07

00−9±10

970/945

0.28

0.20

MS 0015.9+1609

2.59±0.27

085±12

0.43±0.07

0.24±0.07

+070±12

1078/1117

0.79

0.73

MS 0451.6-0305

2.89±0.22

68±8

0.31±0.04

0.26±0.06

+80±8

1193/1117

0.06

0.06

MS 1054.4-0321

2.13±0.22

058±12

0.41±0.10

0.09±0.07

+007±33

1083/1117

0.76

0.73

spherical beta model

Abell 697

2.74±0.24

72±8

0.19±0.02

-

-

1397/1119

0.00

0.00

Abell 1835

2.46±0.31

63±8

0.17±0.02

-

-

1008/947

0.08

0.08

MS 0015.9+1609

2.54±0.29

080±11

0.40±0.05

-

-

1099/1119

0.66

0.61

MS 0451.6-0305

2.92±0.24

53±7

0.24±0.03

-

-

1222/1119

0.02

0.02

MS 1054.4-0321

2.13±0.22

056±10

0.40±0.07

-

-

1085/1119

0.76

0.76

Note. – Table of the best-fit parameters and 1σ
uncertainties for our beta model fits
with the power law exponent β set
to the best-fit value found in P10 (0.86).
From left to right the columns give the normalization
of the beta profile, y0, the core radius of the major axis, rc,
the relative value of the core radius, rc/r500,
the ellipticity, ϵ, the position angle of the
major axis in degrees east of north, θ,
the χ2 and DOF for the fit, and the
goodness of fit quantified by the probability to
exceed the given χ2/DOF based on the
standard χ2 probability distribution function
and also
empirically by the fraction of our 1000
noise realizations that produce a larger
χ2 value when a model cluster is
added to them
(see Section 6.2).
The values of rc/r500 for the spherical fits
can be compared to
the nominal value of 0.20 given in P10
based on spherical fits of SPT SZ data.
See Section 6.2 for a description
of how the parameter uncertainties are calculated.
Note that we have included the 4.3% uncertainty
in our flux calibration in the error estimates for
y0.

Table 3Beta model fit parameters

cluster

y0 (10−4)

β

rc (arcsec)

δRA

δdec

Bolocam

OV/BI/Ch

Bolocam

OV/BI/Ch

Abell 697

3.63±0.31

2.29+0.23−0.24

0.607

49±6

43.2+2.1−2.0

0−6.2±4.2

−14.0±4.6

Abell 1835

2.95±0.37

3.19+0.19−0.21

0.670

49±6

32.4+1.4−1.1

0−8.2±4.4

+01.5±5.0

MS 0015.9+1609

2.80±0.31

2.55+0.15−0.15

0.744

069±10

42.9+2.6−2.4

+18.2±5.2

0+5.3±4.7

MS 0451.6-0305

3.05±0.25

2.72+0.15−0.13

0.795

48±6

36.0+1.9−1.6

+14.2±4.6

0+7.9±4.2

MS 1054.4-0321

1.92±0.20

2.09+0.17−0.17

1.083

070±12

70.5+6.5−6.9

0−1.0±4.4

0−1.5±4.6

Note. – Table of the best-fit parameters and 1σ uncertainties
when we fit a spherical beta model to our data using the value of
β found by B06 using OVRO/BIMA and Chandra data
(in contrast to our nominal beta model
fits with β=0.86).
From left to right the columns give our best-fit values of y0,
the B06 best-fit values of y0, the value of β,
our best-fit values of rc, the B06 best-fit values
of rc, and the RA and dec offsets of the
centroids of our best-fit models
compared to the X-ray centroids of the B06 fits.
Compared to B06, we find a significantly larger
value of y0 for Abell 697, and we
find significantly larger values of rc
for Abell 1835 and MS 0015.9+1609.
We also find small, but measurable, centroid
offsets for all of the clusters other than
MS 1054.4-0321.

4.3. Fitting procedure

Our cluster images are filtered by both the
atmospheric noise subtraction and the
Bolocam point-spread function.
Therefore, prior to fitting a model to our
data, we need to filter the cluster model in an
identical way.
First, we generate a 2-dimensional image
using the cluster model,
computed directly from the isothermal
beta model and via a line-of-sight integration
of the Nagai pressure model.
Next, the model image is convolved with the
Bolocam point-spread function and the
measured transfer function.
In practice, the transfer function convolution
is performed via multiplication in Fourier space.
This filtered model is then compared to our
data map, using all of the map pixels contained
within a radius where the minimum coverage
is greater than 25% of the peak coverage.
This radius is typically between 6 and 7 arcmin.
We use an iterative least-squares technique to
determine the best-fit parameters for each
model;
we estimate the uncertainty on each parameter
via the standard deviation
of the best-fit parameter
values estimated from noise realizations
with model clusters added to them
(see Section 6.2).

For each model, we fit both a scale radius
(rc for the beta model and rs
for the Nagai model) and a normalization
(y0 for the beta model and p0
for the Nagai model).
Additionally, as described above,
we fit for the observationally unconstrained
DC signal level of the map, δT.
We also fit for a centroid offset
relative to the X-ray pointing center.
The offsets for the
five clusters range from ≃0−20±5 arcsec,
indicating there are no major differences
in the X-ray and SZ centroids.
Finally, when we allow the model to be
elliptical in the image plane, we fit
for the ellipticity ϵ and
position angle θ.
A complete list of the best-fit parameters for
spherical and elliptical versions of both models
is given in Tables 2
and 3.

4.4. Discussion and comparison to previous results

All of these clusters have been studied extensively
at a wide range of wavelengths,
providing us with a large number of published
results to compare our model fit parameters to.
In general, our results agree well with
those found from previous studies.
In particular, when we fit a spherical
beta model to our data using the values of
β given in
B06, which were
determined using Chandra
X-ray data and 30 GHz interferometric
SZ data,
our best-fit values
for y0 agree quite well, with
the exception of Abell 697
(see Table 4).
The best-fit values we find for rc
are also consistent with those determined
in B06, with the
exception of Abell 1835 and MS 0451.6-0305.
Additionally, the ellipticity and orientation
of our elliptical fits are in general
consistent
with previously published results
(Maughan et al., 2008; Donahue et al., 2003; De Filippis et al., 2005; Piffaretti et al., 2003; Girardi et al., 2006; McNamara et al., 2006; Schmidt et al., 2001; Neumann and Arnaud, 2000).
Finally, the best-fit concentration
parameters from our Nagai model fits
to three of the five clusters are
consistent with those found from
X-ray data (Arnaud et al., 2009);
MS 0015.9+1609 and MS 1054.4-0321
have significantly lower values of c500.
We describe each cluster in detail below:

Abell 697

We find that Abell 697 has a significant
ellipticity, and it is not well described by either
a beta model or a Nagai model.
The poor model fits result from
the cluster appearing significantly extended in
the SW direction, and extremely compact in the
NE direction.
Abell 697 is the only cluster in our sample
that is not adequately described by the
models.
The ellipticity we find for Abell 697,
ϵ=0.37±0.05, is roughly
consistent
with the ellipticity of ≃0.25 found from X-ray data
(De Filippis et al., 2005; Maughan et al., 2008; Girardi et al., 2006).
We find a position angle of −24 deg for Abell 697,
which is similar to the value of −16 deg found by
Girardi et al. (2006), but somewhat misaligned
to the position angle of 16 deg found
by De Filippis et al. (2005).

Abell 1835

We detect an ellipticity
in our image of
Abell 1835,
and we find that it is well described by
either an elliptical beta or Nagai model.
Our best-fit spherical
Nagai model parameters with A=0.9, B=5.0,
and C=0.4 (p0=11.1±2.6×10−11 erg/cm3,
rs=4.6±1.1 arcmin) are consistent
with the values found in Mroczkowski et al. (2009)
(p0=13.6×10−11 erg/cm3, rs=4.3 arcmin)
using a combination of SZA SZ data and Chandra
X-ray data.
X-ray measurements find an ellipticity of
0.1−0.2 for Abell 1835
(De Filippis et al., 2005; McNamara et al., 2006; Schmidt et al., 2001),
consistent with the value of ϵ=0.27±0.07 we
find with Bolocam.
We find a position angle of −16 deg for Abell 697,
which is similar to the values of
7, −20, and −30 deg found by
De Filippis et al. (2005), McNamara et al. (2006), and
Schmidt et al. (2001).

MS 0015.9+1609

MS 0015.9+1609 appears to be
elliptical in our image,
and it is well described by either a
spherical or elliptical model.
As with Abell 697 and Abell 1835, X-ray data
favor slightly lower ellipticities,
ϵ≲0.20, compared to
what we find with Bolocam,
ϵ=0.24±0.08(De Filippis et al., 2005; Maughan et al., 2008; Piffaretti et al., 2003).
We find a position angle of 68 deg for
MS 0015.9+1609, fairly close to the value
of 47 deg found by Piffaretti et al. (2003),
but almost orthogonal to the value of
−49 deg found by De Filippis et al. (2005).

MS 0451.6-0305

MS 0451.6-0305 also
appears to be elliptical in our image
and is adequately described by either
an elliptical beta or Nagai model.
We find an ellipticity of ϵ=0.26±0.06
for MS 0451.6-0305, in excellent agreement with
ellipticities determined using X-ray data
(De Filippis et al., 2005; Donahue et al., 2003).
Additionally, our best-fit position angle
of 85 deg agrees well with the value
of 84 deg found by De Filippis et al. (2005) and
the value of −75 deg found by Donahue et al. (2003).

MS 1054.4-0321

Our image of MS 1054.4-0321
shows no evidence for ellipticity, and it
is well described by either an elliptical
or spherical model.
Although MS 1054.4-321 does not appear to be
elliptical in our data, X-ray data
show a clear ellipticity oriented
along the east-west direction
(Neumann and Arnaud, 2000; Jeltema et al., 2001).
However, our non-detection of an ellipticity,
ϵ=0.09±0.07, is only
marginally inconsistent
with the X-ray value determined by
Neumann and Arnaud (2000),
ϵ=0.29.

SDS1

We have attempted to fit
cluster models to the SDS1 map, but
we find best-fit amplitudes that are
consistent with 0 and
best-fit scale radii that are
large compared to the size
of our images
(i.e., scale radii that
are large enough to produce
profiles that are approximately
constant over the entire
image).

Rather than using models, which at best provide
an adequate description of what clusters look like
on average, we have chosen to derive observable
quantities from our images in a quasi-model-independent
way.
In Section 3.3 we described how we calculate
a unique transfer function for each cluster to quantify
the effects of our noise filtering.
In the Fourier space of our images,
these transfer functions are a set of
two-dimensional complex numbers that
describe how an input cluster image is filtered
as a function of two-dimensional Fourier mode.
We obtain an unfiltered, or deconvolved, image of the cluster
by Fourier transforming
our image, dividing by the two-dimensional
complex transfer function, and then Fourier transforming
the result back to image space9.
At the largest scales in our map, the transfer function
has a magnitude of ≃0.2
(see Figure 3), resulting in a numerically stable,
but significant, amplification of the large scale
noise.
In particular, the residual atmospheric noise and
primary CMB fluctuations, which had been filtered
to be approximately white, produce a significant
low-frequency noise component in our deconvolved images.
We estimate the noise in our deconvolved images
by deconvolving each of the 1000 noise realizations
for each cluster.
Since the off-diagonal elements of the noise
covariance matrix are significant due to the
low-frequency noise, we estimate all of our measurement
uncertainties from the standard deviation of
measuring the same quantity in each of
our 1000 deconvolved
noise realizations (see below and Section 6.2).
Note that most of the measurement uncertainties
in our processed (i.e., filtered) images
are computed in the same way, even though their
noise covariance matrix is approximately diagonal.

By deconvolving the two-dimensional transfer function
of our data processing we obtain unbiased cluster
images, modulo smoothing with our PSF and the
unconstrained DC signal level.
Although the transfer function was computed using
the best-fit-elliptical Nagai model, and is
therefore somewhat model dependent, the
dependence is negligible other than the
determination of the DC signal offset of the image
(see Section 6.1).
We do not attempt to recover the information
lost due to smoothing by our PSF, but we
use the value of δT found from our
elliptical Nagai model fits to restore the
correct DC signal level to our images.
Due to uncertainties in the model itself,
especially at large radii where there is
little or no observational data, along with
cluster-to-cluster deviations from the model,
this does introduce a non-negligible
model-based bias in our images
(the typical model uncertainty in the DC
signal offset is 5−10μKCMB,
see Section 6.1).
However, we emphasize that this bias only
affects the DC signal level of the images;
the shapes of the cluster profiles are
essentially model-independent.

We have computed a model-independent
value for Y500 from these
deconvolved images, which
is the integrated y within r500
(i.e., the cylindrical Y500
rather than the spherical Y500,
see Table 5).
As mentioned above, since our assumption that the noise covariance
matrix is diagonal fails for the deconvolved
images, we estimate the uncertainty in
our estimate of YSZ via the scatter
among the YSZ values determined from
each of our noise
realizations.
As an example of the amount of low-frequency noise present
in our deconvolved images on the scale of r500, note that
the measurement uncertainty on Y500 is
approximately 10 times larger than it would be
if the noise was white.
For the five clusters in our sample, we are
able to determine YSZ with an
uncertainty of ≃10%,
limited mainly by systematics in determining
the DC signal offset and flux calibration.

We examine the self-similar scaling
that is expected between YSZ and
the total cluster mass using the relation
given in Bonamente et al. (2008),

YSZD2AE(z)−2/3∝f−2/3gasM5/3gas,

where DA is the angular diameter distance,
E(z)=√Ωm(1+z)3+ΩΛ,
and we have assumed Mgas is a good proxy for
the total cluster mass (Allen et al., 2008; Mantz et al., 2010a).
The scatter in this relation is expected to be
≲10% (Kravtsov et al., 2006),
and current measurements are roughly consistent
with this prediction (Morandi et al., 2007; Bonamente et al., 2008; Marrone et al., 2009; Plagge et al., 2010; Huang et al., 2010; Andersson et al., 2010).
Given our small sample, we do not attempt to
constrain the intrinsic scatter in the
Y−M relation, but we follow the
formalism of Marrone et al. (2009) and
P10 to fit a logarithmic
scaling of the form Y=a+bX with the intrinsic
scatter set to 10%.
For the Y500−Mgas,500 relation
we find best-fit values of
a=−5.46±0.84 and b=1.63±0.71,
consistent with the self-similar prediction
of b=5/3.
The overall scatter of our data about the fit
is 13%, consistent with an intrinsic scatter
of ≃10% given our ≃10%
uncertainty on YSZ (see Figure 6).

Additionally, our results are consistent
with YSZ−Mgas scaling
relations measured by other groups.
For example, our results using model-independent
YSZ estimates from Bolocam and Mgas
estimates from Chandra within r500
agree well with the results in P10
using YSZ estimates from the
SPT best-fit beta model and Mgas estimates
primarily from XMM-Newton
(a=−5.73±0.43 and b=2.12±0.45
within r500 and
a=−5.92±0.41 and b=1.97±0.44
within r2500).
The results in Bonamente et al. (2008), using
beta model fits to 30 GHz OVRO/BIMA SZ data
and Chandra X-ray data within
r2500, also match our results quite
well (a=−5.22±1.77 and b=1.41±0.13).

Figure 6.— Scaling relation between YSZ and
Mgas for the five clusters in our
sample.
The black squares give the Bolocam
model-independent cylindrical Y500
estimates
and ChandraMgas,500 estimates
(see Table 1),
and the black dashed line represents the
best fit to these data.
We also show fits of a similar type
obtained by various authors:
red dashed: SPT beta-model-derived
Y500 vs. Mgas,500, primarily
obtained from XMM-Newton data (P10);
solid blue:
same, but within r2500;
dot-dashed green:
OVRO/BIMA beta-model-derived Y2500
vs. Mgas,2500 obtained from
Chandra data (Bonamente et al., 2008).
The difference between the red-dashed
line and our data is likely caused
by systematic differences between
model-derived and model-independent
estimates of Y50010.
The scatter of our data
relative to our best fit is 13%,
consistent with the expected intrinsic
scatter of ≃10% given our
≃10% uncertainty on Y500.

JK noise

CMB/PS

map DC signal

flux cal.

cluster

Y500

σY

σY

σY

σY

Abell 697

59.2±6.3

1.2

0.5

5.6

2.5

Abell 1835

47.5±4.9

1.3

0.5

4.3

2.0

MS 0015.9+1609

23.3±2.2

1.6

0.6

0.9

1.0

MS 0451.6-0305

24.7±2.0

1.2

0.6

0.9

1.1

MS 1054.4-0321

8.8±1.0

0.8

0.4

0.2

0.4

Note. –
Model independent estimates of Y500,
in units of 10−11 ster, along with the
associated uncertainties.
From left to right the columns give the value
and total uncertainty of our estimate of
Y500, the uncertainty due to our
jackknife noise model (i.e.,
excluding CMB and point sources), the
uncertainty due to CMB and point sources,
the uncertainty in the DC signal level of the
map (see Section 6.1),
and the uncertainty in our flux calibration.
Note that the jackknife noise estimate is
dominated by large angular-scale residual
atmospheric fluctuations; the noise
on beam-size scales is negligible.

We are primarily concerned with two types
of systematic errors that may occur in
our analysis:
those caused by our (minimal) use of
a cluster model, specifically an elliptical
Nagai model, and those
caused by characteristics of our noise that
are not properly accounted for in our analysis.
We have run extensive tests to quantify the
level of systematic error we can expect
from each of these two sources.
The details
of these tests are
described below.
In the end, we find that the amount of systematic
error in our images is negligible,
with the exception of the
estimation of the DC signal offset in
our images using the Nagai model.

6.1. Model dependence of results

Since we compute the transfer function for
each cluster using the best-fit elliptical
Nagai model for that cluster,
our deconvolved images necessarily
have some model dependence.
In order to quantify the amount of model
dependence, we have computed transfer
functions for a range of elliptical Nagai
models for one of the clusters in
our sample, MS 0451.6-0305.
Relative to the best-fit model, we have
varied the scale radius, rs, the
ellipticity, ϵ, the position
angle, θ, and the centroid location,
δRA and δdec, by
increasing and decreasing each parameter
individually by its 1σ uncertainty11.
We then deconvolved our processed map of MS 0451.6-0305
using the transfer function computed from
each model and subtracted the resulting map
from the one produced using the transfer
function for the best-fit model.
In each case, the residual map was approximately
white, with an RMS of 1.5, 0.6, and 0.5 μKCMB for
variations in rs and ϵ, θ, and
δRA and δdec, respectively.
Since the typical noise RMS of our deconvolved
maps is ≃10μKCMB, the additional RMS
introduced by our uncertainty in determining
the model used for calculating a transfer
function is quasi-negligible.
Note that the best-fit elliptical Nagai model will not
provide an exact description of a real cluster.
However, the elliptical Nagai model does
provide an adequate description of four of the five
clusters
we have observed, indicating
that the difference between the true cluster
profile and the model profile is in general less than
our noise.
Therefore, the artifacts in our deconvolved map
produced by using a model to describe the
cluster will be smaller than the artifacts
produced by our measurement uncertainty
on the best-fit model.

Additionally, we created a deconvolved map of
MS 0451.6-0305 using the transfer function for a point-like
source.
The resulting profile is significantly different
from the profile obtained using the transfer
function for the best-fit Nagai model,
indicating that the naive calculation of a
transfer function using a point source
is inadequate.
Compared to using the transfer function
calculated from the best-fit elliptical
Nagai model,
the peak decrement is reduced by ≃50μKCMB,
while the magnitude of the
SZ signal at the edge of the map
is increased by ≃50μKCMB (i.e.,
the deconvolved cluster image from a point-source
transfer function is systematically broader).

As described in Section 4, we use the
best-fit elliptical Nagai model to determine the
DC signal level in our maps since its value
is unconstrained by our data.
The measurement uncertainty in the value of
the DC signal is small, <1μKCMB,
but there is a large amount of uncertainty
in the model at large radii.
Based on the results from Borgani et al. (2004),
N07, and Piffaretti and Valdarnini (2008) presented in
Arnaud et al. (2009), the RMS
scatter
in the pressure profiles
from cluster to cluster in simulations is
≲25%.
Therefore, we include an additional 25%
systematic uncertainty on the DC signal
that we add to the deconvolved map.
Specifically, we estimate that the model
uncertainty in the DC signal level
of our map is 25% of the signal
level at the edge of the map.

6.2. Noise characteristics

Figure 7.— Histograms of the χ2 value for 1000
separate noise realizations for MS 0451.6-0305,
overplotted in green with the predicted distribution
assuming the noise covariance matrix is
diagonal (i.e., there is no correlated noise).
For each noise realization the best-fit elliptical
Nagai model profile for MS 0451.6-0305 is added to
the noise realization,
an elliptical Nagai model is fit to this
model-cluster-plus-noise realization,
and the value of
χ2 is computed based on the assumption
that the noise covariance matrix is diagonal.
The vertical red line shows the value of
χ2 for the actual data for MS 0451.6-0305.
The left histogram shows the processed data,
and the right histogram shows the data for
the deconvolved image.
The predicted and actual χ2 distributions
for the processed data overlap, indicating that
there are minimal correlations between map pixels.
However, the actual χ2 distribution for the
deconvolved image data is much broader than the
predicted distribution, indicating that
there are significant noise correlations between
map pixels in the deconvolved images.

As mentioned in Section 3, we make the
approximation that the noise
covariance matrix is diagonal in our processed maps (i.e.,
there are no noise correlations between pixels).
Although fluctuations in both the atmospheric emission
and the CMB are correlated over many pixels, the
high-pass filter we apply to our data timestreams
eliminates these correlations within our ability
to measure them.
To test for noise correlations, we added
processed cluster images of the best-fit
models to
the 1000 noise realizations for each cluster
(i.e., for a given cluster we
separately added each of the four best-fit
cluster models from Tables 2
and 3 to each of the 1000
noise realizations).
We then fit a model of the same type
(e.g., if we added the best-fit
elliptical Nagai model to the noise realization,
then we fit an elliptical Nagai model
to the cluster-model-plus-noise map)
to each of these model-cluster-plus-noise maps
and examined the distribution
of χ2 values for these fits.
For all five clusters,
we found that the distribution of χ2 values
matched the predicted χ2 distribution
obtained from the diagonal noise covariance
matrix12.
As an example, the fit quality of
our measured χ2
distribution for the elliptical-Nagai-model-plus-noise
realizations to the predicted χ2 distribution,
quantified by a PTE,
is 0.58, 0.02, 0.08, 0.66, and 0.21 for
Abell 697, Abell 1835, MS 0015.9+1609,
MS 0451.6-0305, and MS 1054.4+321.
(see Figure 7).
Therefore, we conclude that there is a
negligible amount of correlated noise
in our processed images and our assumption
that the noise covariance matrix is diagonal
is valid.
Furthermore, we have estimated all of
our parameter uncertainties (p0,rs,
etc.) directly from the distribution
of values calculated from our noise realizations.
Consequently, any failure of our assumption
that the covariance matrix is diagonal
for the processed images
will only affect the pixel-weighting
and χ2 values for our model fits.

However, when we perform the same test
of fitting a model to our model-cluster-plus-noise
maps
using noise realizations that have been
deconvolved with our transfer function,
the result is significantly different.
The distribution of χ2 values
calculated from our deconvolved noise
realizations is significantly broader
than the predicted distribution based
on uncorrelated noise
(see Figure 7).
This result is not surprising since the
deconvolution enhances the large-scale
signals in the images, including
residual atmospheric noise and
CMB fluctuations.
Since the noise in the deconvolved images
is significantly spatially correlated, our assumption
that the noise covariance matrix is
diagonal fails.
Therefore, we estimate the uncertainties
for the deconvolved images using the
spread in values for the
noise realizations rather than
from the diagonal elements of the noise covariance
matrix
(e.g., the uncertainties in the radial
profiles are determined from the RMS spread in the
radial profiles of the noise realizations).
This is the same technique used by
Nord et al. (2009) and Basu et al. (2010) to
analyze APEX-SZ data.

The model
fits to the model-cluster-plus-noise realizations
also provide us with estimates of the uncertainties
and biases associated with our model-parameter
fitting.
Specifically, we obtain 1000 best-fit values
for each parameter;
the standard deviation of these values
then gives the uncertainty on our
estimate of that parameter in our
actual data map.
These uncertainties are given in
Tables 2 and
3.
In addition,
if our parameter estimation algorithm is
free from biases, then we should,
on average, recover the parameters of the
input model that was added to the noise
realizations.
In practice, we find a small, but measurable,
bias in our estimates of the pressure
normalization and scale radius in
our fits;
the bias is typically ≲10%
of the uncertainty on each paramter.
We find no measurable bias in our estimates
of the other fit parameters.

Additionally, we have used our signal-free SDS1
maps to further verify our model-fitting proceedure and
to search for any components of the noise that
have not been included in our noise estimate.
First, we
inserted model clusters into the SDS1 data
timestreams based on the
best-fit elliptical Nagai profile for
each of the five clusters in our sample.
These data were then processed and
an elliptical Nagai model was fit to
each resulting image.
In each fit, there are 6 free parameters
(p0, rs, ϵ, θ,
δRA, and δdec), giving
us a total of 30 fit parameters for the
5 model clusters.
Of these 30 fit parameters, 17 (57%) are
within 1σ of the input value,
26 (87%) are within 2σ of the
input value, and all 30 are within
3σ of the input value;
these results
indicate that our model fitting
and parameter error estimation
are working properly.
Additionally, we obtain a reasonable
goodness of fit for the models using
these best-fit parameters,
quantified by a PTE ≃0.8,
providing further evidence
that there is no significant noise
in the data that has not been
included in our noise estimate.
Note that since the five cluster model profiles are
fairly similar, we obtain
comparable PTEs for all five profiles.

We have presented the first results from our program to
image the SZ effect in galaxy clusters with Bolocam.
These images have a beam-smoothed RMS of ≃10μKCMB,
and a resolution of 58 arcsec.
Given this noise level, we are able to measure
SZ signal in radial profiles to approximately the edge of our
maps, which corresponds to 6-7 arcmin
or 1−2 times r500.
In order to subtract noise from atmospheric fluctuations,
we effectively high-pass filter our cluster
images.
However, we are able to deconvolve the effects
of this filter with biases that are negligible
compared to our noise level,
other than our recovery of the DC signal level.
In fitting our images to spherical and elliptical
beta and Nagai models, we find no
preference between the beta and Nagai
models due to the degeneracy between
these models over the angular range
to which our data are sensitive,
but our data do show a definitive
preference for elliptical models
over spherical models.
The weighted mean ellipticity of the five
clusters is ϵ=0.27±0.03,
consistent with results from X-ray data.
Additionally, the best-fit model parameters we determine from
our data are consistent with those found
from previous X-ray and SZ measurements.
We have also obtained model-independent estimates
of YSZ, and we find scaling relations
between YSZ and cluster mass that
are consistent with self-similar predictions,
with a scatter that is consistent with
expectations for a ≃10% intrinsic scatter.

We acknowledge the assistance of:
Nicole Czakon, Tom Downes, and Seth Siegel,
who have provided numerous comments and suggestions
about the data analysis;
the Bolocam instrument team:
P. A. R. Ade, J. E. Aguirre, J. J. Bock, S. F. Edgington,
J. Glenn, A. Goldin, S. R. Golwala,
D. Haig, A. E. Lange, G. T. Laurent,
P. D. Mauskopf, H. T. Nguyen, P. Rossinot, and J. Sayers;
Matt Ferry, who helped collect the data for
Abell 1835 and MS 1054.4-0321;
the day crew and Hilo
staff of the Caltech Submillimeter Observatory, who provided
invaluable assistance during commissioning and data-taking for this
survey data set;
Daisuke Nagai for useful discussions about cluster modeling;
Tony Mroczkowski for providing the best-fit parameters
from his group’s analysis of Abell 1835,
useful comments about cluster modeling,
and pointing out a typo in our original arxiv posting;
the referee for several useful comments and suggestions;
and Kathy Deniston, Barbara Wertz, and Diana Bisel, who provided effective
administrative support at Caltech and in Hilo. Bolocam was constructed and
commissioned using funds from NSF/AST-9618798, NSF/AST-0098737,
NSF/AST-9980846, NSF/AST-0229008, and NSF/AST-0206158. JS
was partially supported by a
NASA Graduate Student Research Fellowship and a NASA
Postdoctoral Program fellowship,
SA and EP were partially supported by NSF grant AST-0649899,
SA was partially supported by the USC WiSE postdoctoral
fellowship and travel grants, and
EP was partially supported by NASA grant NNXO7AH59G
and JPL-Planck subcontract 1290790.

Facilities:CSO.

Appendix A

This appendix includes images and radial profiles of the processed and
deconvolved maps, an image of the
processed residual map after subtracting
the best-fit elliptical Nagai model,
and an image of one of the 1000 noise estimates generated
via jackknife realizations of the data and a model
for the astronomical noise.
We have smoothed all of the images using a Gaussian
beam with a FWHM of 58 arcsec.
A white dot representing the FWHM of the effective
PSF for these beam-smoothed images is given in the
lower left of each image.
The solid white contour lines in the images represent a
S/N of −2,−4,.., and the dashed white contour lines
represent a S/N of +2,+4,....
The deconvolved images contain a significant amount
of noise that is correlated over large angular scales,
along with a model-dependent DC signal offset,
and we therefore do not display noise contours
on the deconvolved images.
The error bars on the radial profiles are estimated
from the spread in radial profiles computed
from our noise realizations, and therefore
do include all of the large-angular-scale noise correlations
(although they do not include the
uncertainty in the DC signal level of the image).
Note that the radial profile bins for the deconvolved
images are correlated due to the large-angular-scale
noise present in those images.

Figure A1.— Abell 697; from left to right and top to bottom we show
the deconvovled image of the cluster,
the processed image of the cluster,
the residual map between the processed image of the
cluster and the best-fit elliptical Nagai model,
one of the 1000 noise realizations for the processed data,
and a binned radial profile.
The contour lines represent a S/N of 2,4,...

Figure A2.— Abell 1835; from left to right and top to bottom we show
the deconvovled image of the cluster,
the processed image of the cluster,
the residual map between the processed image of the
cluster and the best-fit elliptical Nagai model,
one of the 1000 noise realizations for the processed data,
and a binned radial profile.
The contour lines represent a S/N of 2,4,...

Figure A3.— MS 0015.9+1609; from left to right and top to bottom we show
the deconvovled image of the cluster,
the processed image of the cluster,
the residual map between the processed image of the
cluster and the best-fit elliptical Nagai model,
one of the 1000 noise realizations for the processed data,
and a binned radial profile.
The contour lines represent a S/N of 2,4,...

Figure A4.— MS 0451.6-0305; from left to right and top to bottom we show
the deconvovled image of the cluster,
the processed image of the cluster,
the residual map between the processed image of the
cluster and the best-fit elliptical Nagai model,
one of the 1000 noise realizations for the processed data,
and a binned radial profile.
The contour lines represent a S/N of 2,4,...

Figure A5.— MS 1054.4-0321; from left to right and top to bottom we show
the deconvovled image of the cluster,
the processed image of the cluster,
the residual map between the processed image of the
cluster and the best-fit elliptical Nagai model,
one of the 1000 noise realizations for the processed data,
and a binned radial profile.
The contour lines represent a S/N of 2,4,...

Figure A6.— SDS1; from left to right and top to bottom we show
the processed image of the field,
one of the 1000 noise realizations for the processed data,
and a binned radial profile.
The contour lines represent a S/N of 2,4,...
The thin grey lines show the radial profiles for each
of the 1000 noise realizations.

Footnotes

affiliation: California Institute of Technology, Pasadena, CA 91125

affiliation: jack@caltech.edu

affiliation: California Institute of Technology, Pasadena, CA 91125

affiliation: University of Southern California, Los Angeles, CA 90089

affiliation: University of Southern California, Los Angeles, CA 90089

We show that there is no measurable
fixed-pattern or scan-synchronous noise
in our data later in this section
and in Section 6.2.

This approximation is justified for
our processed data maps in Section 6.2.
Note that the approximation fails
for our deconvolved images
(see Section 5), which contain
a non-negligible amount of correlated
noise.
We describe how this correlated noise
is accounted for in our results in Section 5.

As P10 point out,
this value for β is larger than those
generally found from X-ray data,
in agreement with simulations (Hallman et al., 2007).
Additionally, it suggests that the ICM
temperature is falling with increasing
radius, in agreement with simulations and
data (e.g., N07).

This method can be compared to the deconvolution method
employed by APEX-SZ for their analysis of
Abell 2163 and Abell 2204 (Nord et al., 2009; Basu et al., 2010).
They first determine what a point-like object looks
like in their image after being filtered.
Next, they fit this filtered point-source image
to the map pixel with the largest S/N and
subtract it from the image.
The process is repeated until the map is
consistent with noise;
the sum of all the unfiltered point-like images
removed from the map
gives the deconvolved cluster image.

P10 calculated both model-derived
and model-independent estimates of
YSZ for the 15 clusters
in their sample.
Within a radius of r2500,
the two estimates of Y2500
are on average consistent with
each other.
However, the P10 model-derived estimates
of Y500 are on average a factor
of 1.5 larger than the model-independent
estimates for the 10 clusters used
to determine the YSZ−Mgas
scaling relation.
As a result, the P10 model-derived scaling
relation for Y500−Mgas,500 will
be systematically higher by log10(1.5)≃0.2
compared to a model-independent scaling
relation.

The power law slopes (A, B,
and C) were held fixed for all
of our model fits. Due to the large
degeneracy between these values and
rs, we have effectively included
variations in the power law slopes
by varying the value of rs.

We have fit for Nparams free parameters
in our fits to both the actual data and the
noise realizations,
with Nparams=5 or 7 for the
spherical or elliptical fits.
Therefore, the predicted χ2
distribution is for Npix−Nparams
DOF, where Npix is the number of map
pixels.

We show that there is no measurable
fixed-pattern or scan-synchronous noise
in our data later in this section
and in Section 6.2.

This approximation is justified for
our processed data maps in Section 6.2.
Note that the approximation fails
for our deconvolved images
(see Section 5), which contain
a non-negligible amount of correlated
noise.
We describe how this correlated noise
is accounted for in our results in Section 5.

As P10 point out,
this value for β is larger than those
generally found from X-ray data,
in agreement with simulations (Hallman et al., 2007).
Additionally, it suggests that the ICM
temperature is falling with increasing
radius, in agreement with simulations and
data (e.g., N07).

This method can be compared to the deconvolution method
employed by APEX-SZ for their analysis of
Abell 2163 and Abell 2204 (Nord et al., 2009; Basu et al., 2010).
They first determine what a point-like object looks
like in their image after being filtered.
Next, they fit this filtered point-source image
to the map pixel with the largest S/N and
subtract it from the image.
The process is repeated until the map is
consistent with noise;
the sum of all the unfiltered point-like images
removed from the map
gives the deconvolved cluster image.

P10 calculated both model-derived
and model-independent estimates of
YSZ for the 15 clusters
in their sample.
Within a radius of r2500,
the two estimates of Y2500
are on average consistent with
each other.
However, the P10 model-derived estimates
of Y500 are on average a factor
of 1.5 larger than the model-independent
estimates for the 10 clusters used
to determine the YSZ−Mgas
scaling relation.
As a result, the P10 model-derived scaling
relation for Y500−Mgas,500 will
be systematically higher by log10(1.5)≃0.2
compared to a model-independent scaling
relation.

The power law slopes (A, B,
and C) were held fixed for all
of our model fits. Due to the large
degeneracy between these values and
rs, we have effectively included
variations in the power law slopes
by varying the value of rs.

We have fit for Nparams free parameters
in our fits to both the actual data and the
noise realizations,
with Nparams=5 or 7 for the
spherical or elliptical fits.
Therefore, the predicted χ2
distribution is for Npix−Nparams
DOF, where Npix is the number of map
pixels.